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For a graph $G$, let $S_2(G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and $f(G)=e(G)+3-S_2(G)$. Oliveira, Lima, Rama and Carvalho conjectured that $K^+_{1,n-1}$ (the star graph with an additional edge) is the unique graph with minimum value of $f(G)$ on $n$ vertices. In this paper, we prove this conjecture, which also confirm a conjecture for the upper bound of $S_2(G)$ proposed by Ashraf et al., Comment: 15 pages, 5 figures

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. Akbari et al. \cite{S. Akbari} proved that for a complete multipartite graph $K_{t_1 ,\ldots,t_k}$, if $t_i\geq 2 \ (i=1,\ldots,k)$, then deleting any edge will increase the energy. A natural question is how the energy changes when $\min\{t_1 ,\ldots,t_k\}=1$. In this paper, we will answer this question and completely determine how the energy of a complete multipartite graph changes when one edge is removed.

The energy of a graph is defined as the sum of the absolute values of the eigenvalues of the graph. In this paper, we present a new method to compare the energies of two $k$-subdivision bipartite graphs on some cut edges. As the applications of this new method, we determine the first $\lfloor\frac{n-7}{2}\rfloor$ largest energy trees of order $n$ for $n\ge 31$, and we also give a simplified proof of the conjecture on the fourth maximal energy tree.